= Student solutions for Assignment #3 = Solution Sample ---- == Problem 50 == *Problem 50 - Tan Dang == Problem 73 == Show that if p is a prime such that there is an integer b with p = b2 + 4, then $ \mathbb{Z}[\sqrt{p}] $ is not a unique factorization domain. *rhea/images/f/f6/Problem_73_Zeller.pdf
*Solution by Avi Steiner ::My solution only uses the fact that p is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC) == Problem 94 == Show f(x) = x4 + 5x2 + 3x + 2 is irreducible over the field of rational numbers. *Solution by Nicole_Rutt == Problem 101 == (a) Show that x4 + x3 + x2 + x + 1 is irreducible in $ \mathbb{Z}_3[x] $. (b) Show that x4 + 1 is not irreducible in $ \mathbb{Z}_3[x] $ *Solution == Problem 107 == Let R be a commutative ring with identity such that the identity map is the only ring automorphism of R. Prove that the set N of all nilpotent elements of R is an ideal of R *Solution by Avi Steiner == Problem 114 == A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal. *Solution by Nathan Moses
---- Back to 2013 Summer MA 598A Weigel
